precip and sst

Jie He

2. Field1 means and variances: space, time, spacetime

1. The GRAND (space+time) mean of my x and y: 4.7379mm/d and 27.5743Co .
2. The GRAND (space+time) variance of my x and y: 12.6279(mm/d)^2 and 2.9326 (Co)^2 The GRAND standard deviations are: 3.5536mm/d and 1.7125 Co
3. The SPATIAL variance of my TIME MEAN longitude section is: 5.7562(mm/d)^2 and 1.8427(Co)^2.
4. The TEMPORAL variance of my LONGITUDE MEAN time series is: 0.9405(mm/d)^2 and 0.3616(Co)^2.

3. (the assignment part)

1. Confirm that the time mean of the anomalies as defined above is 0. Yes
2. Is the spatial mean of the anomalies (as defined above) 0?
   • No
   • Is it the same as the time series of the spatial mean of the raw data? Or is it a new thing?
     • It’s a new thing.

1. My CLIMATOLOGICAL ANNUAL CYCLE have variance:
   • var( climy12(:),1) 7.9339 (precip)
   • var( climy12(:),1) 2.4852 (sst)
2. My INTERANNUAL ANOMALY ARRAYS have variance:
   • var( anomx(:),1) 4.6939 (precip)
     var( anomy(:),1) 0.4473 (sst)
3. Fill out a variance decomposition table for field 1: feel free to add columns if you can define other parts.

	precip	sst
a) total variance of x	12.6279	2.9326
b) purely spatial (variance of TIME mean at each lon)	5.7562	1.8427
c) variance of (x minus its TIME mean at each lon)	6.8716	1.0898
d) purely temporal (variance of LON mean at each time)	0.9405	0.3616
e) variance of (x minus its LON mean at each time)	11.6873	2.5710
f) remove both means (space-time variability)	12.6279	2.9326
g) mean seasonal cycle	7.9339	2.4852
h) deseasonalized anomalies	4.6939	0.4473
i) variance of longitudinal mean of h	0.2475	0.1025
j) h minus i	4.4464	0.3448

6. Discuss your results:

• (a) = (g) + (h).
  precip: 12.6279=7.9339+4.6939
  sst: 2.9326=2.4852+0.4473

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4. Further decomposition of anomx by scale (using rebinning).

Characteristic time scale : 1mon for Precip; 10mon for SST.
Characteristic space scale : 45o for Precip and SST.

5. Scatter plot, correlation and covariance, regression-explained variance

1. Based on your data fields (which you've seen pictures of), make subsets of your 2 variables x and y and make a scatter plot of these showing the strongest (positive or negative) correlation of one field with the other you can find. The subset might simply be all (x,t) values if your fields are very similar (olr, precip), or maybe the 240 time values at one longitude, or 144 longitudinal values in the time mean, or time series at different longitudes if some variability is offset in your two fields (like pressure and wind).
   • After calculating corrcoef(precip,sst) at different longitudes, I found that the highest correlation coefficient was found at longitude 267.5°

• . The scatter plot for these two variables at longitude 267.5° is:

1. Now consider the covariance and correlation of the two subset arrays entering your scatterplot.
   • What is the correlation coefficient corresponding to this scatter plot? rho = corrcoef(x(:),y(:)) rho = correlate(x,y)
     • rho =
       1.0000 0.8687
       0.8687 1.0000
   • What are the standard deviations of your two data subsets? std(x(:))stdev(y)
     • pcm_std =
       2.0476
     • scm_std =
       1.8924
   • What fraction of the variance of y can be 'explained' by linear regression on x (y = mx + b)? How does this relate to rho? How much y variance is explained? (variance: with units of y squared) What is m? Hint: these are simple questions: use the math formula, not a computer code (Hsieh section 1.4.2, Eq. 1.33).
     • The fraction of the variance of sst can be 'explained' by linear regression on precip is rho.^2=0.7546.
     • The explained variance of sst is scm_std^2*rho.^2=2.7024(Co)^2.
     • m is the slope of the least-squares fit line rho*scm_std/pcm_std=0.8029.
   • What fraction of the variance of x can be 'explained' by linear regression on variable y? (x = nx + a)? How does this relate to rho? What is n? Hint: these are simple questions, use the math formula not computer code.
2. Now add uncorrelated (random) noise with variance 1 to one of your variables. This might be like observation error. noisey = y + random('Normal',0,1,size(y))
   • How did the variance of y change when this noise was added? var(y(:)) var(noisey(:))
     • My original sst variance @ 267.5° was 3.5813 (Co)^2. The variance of my sst + random noise is: 4.1044(Co)^2. The addition of noise increases the variance.
   • How did the correlation change? rho = corrcoef(x(:),noisey(:)) rho = correlate(x, noisey)
     • The new rho is: rho = 1.0000 0.7598 0.7598 1.0000
   • How do these changes affect the regression of y on x? How much (y+noise) variance is explained by linear regression on x? What is the new value of m in the new (noisey = mx + b) regression?
     • The fraction of the variance of sst can be 'explained' by linear regression on precip+noise is rho.^2=0.5773.
     • The explained variance of sst is scm_std^2*rho.^2=2.0674(Co)^2.
     • m is the slope of the least-squares fit line rho*scm_std/pcm_std=0.7022

6. Lagged correlation, covariance, and cross-covariance: questions

1. Show the zero-lag spatial covariance and correlation structures for your primary field, like [[file/view/OLR_anoms_covar_correl.BEM.Matlab.png|OLR_anoms_covar_correl.BEM.Matlab.png]] this for OLR. (please label the axes better than I did!) Interpret the results.
   • 
   • The largest covariance and correlation can be seen between longitudes 200-275. This is where the ENSO signal is the strongest.
2. Show longitude-lag sections of the covariance or correlation of this field, for a base point at some longitude of interest. Like this for OLR at a central Pacific longitude: [[file/view/OLR.lagregression.BEM.jpg|OLR.lagregression.BEM.jpg]] (Please label the axes better than I did in this example! I hate Matlab). Better in IDL: [[file/view/olr_lag_covariances.gif|olr_lag_covariances.gif]]
3. Intepret the results in terms of the characteristic space and time scales of your anomalies. Can you see these characteristic scales in your original raw data like in [[file/view/olr_lag_covariances.gif|olr_lag_covariances.gif]]?
4. 
   
   • The characteristic space and time scales found in question 4 were 45° and 1 month. The contour associated with covariances above approximately 0.9171 corresponds to this space and time scale.
     
     
     
5. Share a longitude-lag slice of your lagged co-variance matrix for your TWO fields. Label it, interpret it.
   • Similar to the SST covariance lag-lon plot.